formula for probability experiment

Since your main problem seems to be how to properly define the sample space and event space this is what i will focus on.

First of all it is worth noting that there is no unique way of defining the sample space, but a reasonable sample space to consider would be the set $Omega$ consisting of all ways to shuffle a set of 52 cards. That is $Omega$ has $52!$ elements and the elements are on the form $omega = (text{Ace of clubs}, text{4 of spades}, text{Queen of hearts},dots)$. Now events are by definition subsets of the sample space, which means that there is a total of $2^{52!}$ events in total.

It is fair to assume that any outcome in $Omega$ has the same probability, so the probability measure we are considering is the probability measure $$mathbb{P}(A) = frac{|A|}{52!}quad text{for $Asubseteq Omega$}.$$ Where $|A|$ is number of elements in $|A|$. So calculating probabilities has been reduced to a counting problem. For instance if we want to compute the probability of the event $$A = {omega in Omega : | : text{The first two cards of $omega$ have the same rank}}$$ we would need to count the number of ways we can shuffle a set of cards such that first two cards have the same rank. Since there are $52$ possibilities for the first card and only $3$ for the second card and then $50!$ for the remaining cards, we get that $$|A| = 52cdot 3 cdot 50!$$ and thus we get that $$mathbb{P}(text{The first two cards have same rank})=frac{|A|}{52!} = frac{3}{51} = frac{1}{17}$$ I hope this helps in understanding the problem and gives you an idea of how to compute the remaining $50$ cases.